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Abstract

We present a Finite Element Method (FEM) to calculate the complex valued k(ω) dispersion curves of a photonic crystal slab in presence of both dispersive and lossy materials. In particular the method can be exploited to study plasmonic crystal slabs. We adopt Perfectly Matched Layers (PMLs) in order to truncate the open boundaries of the model, including their related anisotropic permittivity and permeability tensors in the weak form of Helmholtz's eigenvalue equation. Results of the model are presented in the interesting case of a holey metal film enabling to study the observed extraordinary optical transmission properties in term of the plasmonic Bloch modes of the structure.

(a) Reflectance map compared with the calculated dispersion curves (black lines), green dashed line is the light line; (b) Comparison between bands obtained with (black) and without (red) PML domains (in both cases the tolerance parameter κ was set to 0.1, and periodicity in z direction is 5μm).

(a) z-component of the magnetic field obtained by exciting the proper mode by mean of excitation boundary conditions. Frequency is ω = 3.5∙1015Hz, d = 600nm and a = 50nm. (b) Comparison between the imaginary part of the modes calculated with illumination (blue curve) and by modal analysis (magenta curve) respectively at fixed amplitude a = 50nm.

Maps of TM and TE transmittance through arrays of square nano-holes (period d = 940nm) in a 200nm thick Silver film in air compared with the calculated Bloch-modes dispersion curves (black lines, labeled as (|m|,|n|) ± ); (a) and (b) refer to holes with size a = 250 nm, (c) and (d) to a = 500nm; (e) and (f) are zooms of the regions marked with 1 and 2 respectively in (a). Black dashed line marks the light line, white and red solid lines mark the flat SPP dispersions, ( ± 1,0) and (0, ± 1) respectively. Figure 6(f) contains also a comparison between the x-component of the magnetic field profile in the z-y plane obtained with modal analysis and illumination respectively.

(a) Hy and (b) |Ex| fields of the TM modes found at frequency of 1.6∙1015Hz for a = 250nm (I,II) and a = 500nm (III,IV). The calculated eigenvalues in the four cases are respectively kx/(π/d) = 0.3828 + 0.0019i, 0.3878 + 0.0011i, 0.2792 + 0.1251i, 0.3829 + 0.0025i. Modes are classified as antisymmetric (I,III) and symmetric (II,IV) according to the symmetry of Hy field with respect to the z = 0 plane. PML size is fixed at 2d for convenience.

Real and imaginary parts of the modes for a = 250nm (a,b) and a = 500nm (c,d). Blue and red dots represent the TM (1,0)- and TM (1,0)+ modes respectively, black dots represent the TE and TM (0,1) ± modes. Insets in (a) report the dominant magnetic field component profile (colorscale) and the (H) field (arrows) respectively in a x-y plane laying 10 nm above the slab at frequency of 2.05x1015Hz for the (|m|,|n|)- modes.